Need a confirmation Let's say I have a system of inequalities { A x ≤ b x ≥ 0 A ∈ R m × n , x ∈ R n × 1 , b ∈ R m × 1 . n ≥ m Assume also that A has full rank. Is it possible to manipulates the inequalities involved such that for i = 1 … mI have that x i is bounded (above or below) by x i − 1 , . . . , x 0 ? Something like a i i x i ≤ C i + ∑ j = 0 i − 1 c i j x j

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Need a confirmation Let&#039;s say I have a system of inequalities      {                            A          x          ≤          b                                      x          ≥          0                          A  ∈        R          m      ×      n      ,   x  ∈        R          n      ×      1      ,   b  ∈        R          m      ×      1      .   n  ≥  m Assume also that   A has full rank. Is it possible to manipulates the inequalities involved such that for   i  =  1  …  mI have that       x    i   is bounded (above or below) by       x          i      −      1        ,  .  .  .  ,      x    0  ? Something like      a          i      i            x    i    ≤      C    i    +      ∑          j      =      0              i      −      1            c          i      j            x    j

jarakapak7 · Accepted Answer

The form you advocate,       a          i      i            x    i    ≤      ∑          j      =      0              i      −      1            c          i      j            x    j  has no constants anymore. Now this is not going to be the general case. As a contradictive example, consider the simple special case where the matrix A is diagonal. Then the bounds just read      a          i      i            x    i    ≤      b    i  which contradicts your form.Besides that, there is also a contructive answer to your question. Your system of inequalities      {                            A          x          ≤          b                                      x          ≥          0                      }  actually describe the feasible region of a simplex and you will find abundant literature on that, e.g. here. From the point of view of your question, the feasible region is a convex polytope which can always be described, using slack variables, in standard form as      {                                                A            ~                                        x            ~                    =                              b            ~                                                                    x            ~                    ≥          0                      }  So this answers your question (if you include slack variables) since with this theorem, you have the relation       A    ~        x    ~    =      b    ~   between the variables.

Need a confirmation Let's say I have a system of inequalities { <mtable columnalign=

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